Summary of five knowledge points of compulsory mathematics in senior one 1
Basic properties of difference sequences
Arithmetic progression with a tolerance of (1) is still arithmetic progression, and its tolerance is still D. 。
⑵ For arithmetic progression whose tolerance is d, the sequence obtained by multiplying each term by the constant k is still arithmetic progression, and its tolerance is kd.
(3) If {a} and {b} are arithmetic progression, {a b} and {ka+b}(k and b are nonzero constants) are also arithmetic progression.
(4) For any m and n, arithmetic progression {a} has: a=a+(n-m)d, especially when m= 1, arithmetic progression's general formula is obtained, which is more general than arithmetic progression's general formula.
5. Generally speaking, if L, K, P, …, M, N, R, … are all natural numbers, l+k+p+…=m+n+r+… (the number of natural numbers on both sides is equal), then when {a} is arithmetic progression, there is: A+A+.
[6] arithmetic progression with a tolerance of d, from which equidistant terms are extracted, forms a new series, which is still arithmetic progression, and its tolerance is kd(k is the difference between the number of extracted terms).
(7) If {a} is a arithmetic progression with a tolerance of d, then A, A, …, A and A are also arithmetic progression with a tolerance of -d; In arithmetic progression {a}, a-a=a-a=md (where m, k,).
In arithmetic progression, from the first term, every term (except the last term of a finite series) is the arithmetic average of the two terms before and after it.
Levies when the tolerance d >. 0, the number in arithmetic progression increases with the increase of the number of terms; When d < 0, the number in arithmetic progression decreases with the decrease of the number of terms; When d=0, the number in arithmetic progression is equal to a constant.
⑽ Let A, A and A be three terms in arithmetic progression, and the ratio of the distance difference between A and A, A and a=(≦- 1), then A =.
(1) The necessary and sufficient condition for the sequence {a} to be arithmetic progression is that the sum of the first n terms of the sequence {a} can be written in the form of S=an+bn (where a and b are constants).
(2) In arithmetic progression {a}, when the number of terms is 2n(nN), S-S=nd, =; When the number of terms is (2n- 1)(n), S-S=a, =.
(3) If the sequence {a} is arithmetic progression, then S, S-S, S-S, ... are still arithmetic progression with an error of.
(4) If the sum of the first n terms of two arithmetic progression {a} and {b} is s and t respectively (n is odd), then =.
5] In arithmetic progression {a}, S=a, s = b (n >; M), then S=(a-b).
[6] In arithmetic progression {a}, it is a linear function of n, and all points (n,) are on the straight line y=x+(a-).
(7) Remember that the sum of the first n items of arithmetic progression {a} is S.① If a >；; 0, tolerance d
Basic properties of geometric series
(1) geometric progression whose common ratio is q, from which equidistant terms are taken out to form a new series, which is still geometric progression and whose common ratio is q(m is the difference of equidistant terms).
⑵ For any m and n, in the geometric series {a}, there are: a = a q, especially when m= 1, the general term formula of geometric series is obtained, which is more universal than that of geometric series.
(3) Generally speaking, if t, k, p, …, m, n, r, … are all natural numbers, and t+k, p, …, m+…=m+n+r+… (the number of natural numbers on both sides is equal), then when {a} is a geometric series, there are: a.a. ..
(4) If {a} is a geometric series whose common ratio is q, {|a|}, {a}, {ka} and {} are also geometric series, and their common ratios are |q|}, {q}, {q} and {}.
5] If {a} is a geometric series and the common ratio is q, then a, a, a, …, a, … are geometric series with q as the common ratio.
[6] If {a} is a geometric series, then for any n, there is a = a q >；; 0.
(7) The series formed by the product of two geometric progression counterparts is still geometric progression, and the common ratio is equal to the product of the common ratio of these two series.
By q> 1 and a>0 or 00 and 0 1, the geometric series is a decreasing sequence; When q= 1, geometric progression; When asked
Senior high school mathematics compulsory five: the first n terms of geometric series and the basic properties of formula S.
(1) If the series {a} is a geometric series with a common ratio of q, then the sum of the first n terms is S=
That is to say, the sum of the first n terms of the geometric series whose common ratio is q is a series of function values of the piecewise function of q, and the boundary of the piecewise function is q= 1. Therefore, it is necessary to find out whether the common ratio q may be equal to 1 by using the formula of the first n terms and geometric series. If q may be equal to 1, it is necessary to divide it by Q = 1.
(2) When a, q and n are known, the formula S =；; When a, q and a are known, the formula S= is used.
(3) If S is a geometric series whose common ratio is q, then S = S+QS. (2).
(4) If the sequence {a} is geometric progression, then S, S-S, S-S, ... still become geometric series.
5. If the geometric series of 3n terms (q≦- 1) has the first n-term sum and the first n-term product s and t, and the second n-term sum and the second n-term product s and t, respectively, then s, s and s are geometric series, and t and t are also geometric series.
Universal formula: sin2α = 2tanα/( 1+tan 2α) (note: tan 2α refers to tan squared α).
cos2α=( 1-tan^2α)/( 1+tan^2α)tan2α=2tanα/( 1-tan^2α)
Power lifting formula:1+cos α = 2cos2 (α/2)1-cos α = 2sin2 (α/2)1sin α = (sin (α/2) cos (α/2)) 2.
Power-off formula: cos2α = (1+cos2α)/2sin2α = (1-cos2α)/21) sin (2kπ+α) = sinα, cos (2kπ+α) = cos α, tan (.
(2)sin(-α)=-sinα，cos(-α)=cosα，tan(-α)=-tanα，cot(-α)=-cotα
(3)sin(π+α)=-sinα，cos(π+α)=-cosα，tan(π+α)=tanα，cot(π+α)=cotα
(4)sin(π-α)=sinα，cos(π-α)=-cosα，tan(π-α)=-tanα，cot(π-α)=-cotα
(5)sin(π/2-α)=cosα，cos(π/2-α)=sinα，tan(π/2-α)=cotα，cot(π/2-α)=tanα
(6)sin(π/2+α)=cosα，cos(π/2+α)=-sinα，
tan(π/2+α)=-cotα，cot(π/2+α)=-tanα
(7)sin(3π/2+α)=-cosα，cos(3π/2+α)=sinα，
tan(3π/2+α)=-cotα，cot(3π/2+α)=-tanα
(8)sin(3π/2-α)=-cosα，cos(3π/2-α)=-sinα，
Tan (3π/2-α) = COT α, COT (3π/2-α) = Tan α (k π/2 α), where k ∈ z.
Note: for the convenience of doing the problem, we used to regard α as an angle located in the first quadrant and less than 90;
When k is an odd number, the trigonometric function on the right side of the equation changes, for example, sin becomes cos. Even numbers remain unchanged;
Use the quadrant of the angle (k π/2 α) to determine the sign of the trigonometric function on the right side of the equation. Example: tan(3π/2+α)=-cotα.
In this formula, k=3 is an odd number, so the strain on the right side of the equation is cot.
Moreover, the angle ∵ (3π/2+α) is in the fourth quadrant, and tan is negative in the fourth quadrant, so in order to make the equation hold, the right side of the equation should be -cotα. The positive and negative distribution of trigonometric function in each quadrant.
Sin: the first and second quadrants are positive; Third and fourth quadrant negative cos: first and fourth quadrant positive; Cot and tan in the second and third quadrants are negative: the first and third quadrants are positive; The second and fourth quadrants are negative.
Summary of five knowledge points of compulsory mathematics in senior high school
(A), mapping, function, inverse function
The concepts of 1, correspondence, mapping and function are both different. Mapping is a special correspondence, and function is a special mapping.
2, for the concept of function, should pay attention to the following points:
(1) Knowing the three elements of a function can determine whether two functions are the same.
(2) Master three representations-list method, analytical method and mirror method, and seek the functional relationship between variables according to practical problems, especially the analytical formula of piecewise function.
(3) If y=f(u) and u=g(x), then y=f[g(x)] is called a composite function of f and g, where g(x) is an inner function and f(u) is an outer function.
3. The general steps to find the inverse function of the function y=f(x):
(1) Determine the range of the original function, that is, the definition range of the inverse function;
(2) x = f-1(y) is obtained from the analytical formula of y=f(x);
(3) Exchange x and y to get the idiomatic expression y=f- 1(x) of the inverse function, and mark the domain.
Note ①: For the inverse function of piecewise function, first find the inverse function on each segment separately, and then merge them together.
② Be familiar with the application, find the value of f- 1(x0), and make rational use of this conclusion to avoid the process of finding the inverse function, thus simplifying the operation.
(2), the analytical formula and definition of the function
1, the function and its domain are an inseparable whole, and the function without domain does not exist. Therefore, in order to correctly write the analytical expression of the function, we must find the corresponding law between variables and the definition domain of the function. There are usually three ways to find the function domain:
(1) Sometimes a function comes from a practical problem, so the independent variable x has practical significance, and the domain should be considered in combination with the practical significance;
(2) Find the domain of the analytic formula of the known function, as long as the analytic formula is meaningful, such as:
The denominator of (1) score must not be zero;
(2) The number of even roots is not less than zero;
③ The true value of logarithmic function must be greater than zero;
④ The bases of exponential function and logarithmic function must be greater than zero and not equal to1;
⑤ tangent function y=tanx(x∈R and k∈Z) and cotangent function y=cotx(x∈R, x ? k π, k∈Z) in trigonometric functions.
It should be noted that when the analytic expression of a function consists of several parts, the domain is the common part (i.e. intersection) of each meaningful independent variable.
(3) Knowing the domain of one function and finding the domain of another function mainly consider the profound meaning of the domain.
It is known that the domain of f(x) is [a, b], the domain of f[g(x)] means that x satisfies the range of a≤g(x)≤b, and the domain of f[g(x)] is known as x ∈ [a, b].
2. Generally speaking, there are four ways to find the analytic expression of a function.
(1) When it is necessary to establish a function relationship according to practical problems, it is necessary to introduce appropriate variables and find the analytical formula of the function according to the relevant knowledge of mathematics.
(2) Sometimes, given the characteristics of a function, we can use the undetermined coefficient method to find the analytical expression of the function. For example, if the function is linear, let f(x)=ax+b(a≠0), where a and b are undetermined coefficients. According to the conditions of the problem, list the equations and find out a and B.
(3) If the expression of the compound function f[g(x)] is given, method of substitution can be used to find the expression of the function f(x), and then the range of g(x) must be found, which is equivalent to finding the domain of the function.
(4) If it is known that f(x) satisfies an equation, and other unknowns (such as f(-x), etc. ) Except for f(x), all other equations appearing in this equation must be constructed according to the known equation, and the expression of f(x) can be obtained by solving the equation.
(3) The range and maximum value of the function
The range of 1. function depends on the defined range and the corresponding law. No matter what method is used to find the function range, we should first consider defining the range. The common methods to find the range of functions are as follows:
(1) direct method: also known as observation method, for functions with simple structure, the range of the function can be directly observed by applying the properties of inequality to the analytical expression of the function.
(2) Substitution method: A given complex variable function is transformed into another simple function re-evaluation domain by algebraic or trigonometric substitution. If the resolution function contains a radical, algebraic substitution is used when the radical is linear and trigonometric substitution is used when the radical is quadratic.
(3) Inverse function method: By using the relationship between the definition domain and the value domain of the function f(x) and its inverse function f- 1(x), the value domain of the original function can be obtained by solving the definition domain of the inverse function, and the function value domain with the shape of (a≠0) can be obtained by this method.
(4) Matching method: For the range problem of quadratic function or function related to quadratic function, the matching method can be considered.
(5) Evaluation range of inequality method: Using the basic inequality a+b≥[a, b∈(0, +∞)], we can find the range of some functions, but we should pay attention to the condition of "one positive, two definite, three phases, etc." Sometimes you need skills such as Fang.
(6) Discriminant method: y=f(x) is transformed into a quadratic equation about x, and the definition domain is evaluated by "△≥0". The characteristic of the question type is that the analytical formula contains roots or fractions.
(7) Finding the domain by using the monotonicity of the function: When the monotonicity of the function on its domain (or a subset of the domain) can be determined, the range of the function can be found by using the monotonicity method.
(8) Finding the range of function by combining numbers and shapes: using the geometric meaning expressed by the function, finding the range of function by geometric methods or images, that is, finding the range of function by combining numbers and shapes.
2. Find the difference and connection between the maximum value of the function and the range.
The common method of finding the maximum value of a function is basically the same as the method of finding the function value domain. In fact, if there is a minimum (maximum) number in the range of a function, this number is the minimum (maximum) value of the function. Therefore, the essence of finding the maximum value of a function is the same as that of the evaluation domain, but the angle of asking questions is different, so the way of answering questions is different.
For example, the value range of the function is (0, 16), and the value is 16, and there is no minimum value. For example, the range of the function is (-∞, -2]∩[2,+∞), but this function has no value and minimum value, only after changing the definition of the function, such as X >；; 0, and the minimum value of the function is 2. The influence of definition domain on the range or maximum value of a function can be seen.
3. The application of maximum function in practical problems.
The application of function maximum is mainly reflected in solving practical problems with function knowledge, which is often expressed in words as "minimum project cost", "profit" or "minimum area (volume)" and many other practical problems. When solving, we should pay special attention to the restriction of practical significance on independent variables in order to get the maximum value correctly.
(4) Parity of functions
1. Definition of function parity: For function f(x), if any x in the function definition domain has f(-x)=-f(x) (or f(-x)=f(x)), then function f(x) is called odd function (or even function).
To correctly understand the definitions of odd function and even functions, we should pay attention to two points: the symmetry of the domain on the (1) number axis is a necessary and sufficient condition for the function f(x) to be a odd function or even function; (2)f(x)=-f(x) or f(-x)=f(x) is a unit element in the domain. (Parity is a global property over the domain of a function. ) 。
2. The definition of parity function is the main basis for judging the parity of function. In order to judge the parity of a function, it is sometimes necessary to simplify the equivalent form of the function or application definition:
Pay attention to the application of the following conclusions:
(1) f(|x|) is always an even function whether f(x) is a odd function or an even function;
(2)f(x) and g(x) are odd function in the fields D 1 and D2, respectively, so on D 1∩D2, f(x)+g(x) is odd function, F (x) G (x) is an even function, and similarly ".
(3) The parity of the compound function of the even-odd function is usually an even function;
(4) The derivative function of odd function is even function, and the derivative function of even function is odd function.
3. Some properties and conclusions about parity.
The necessary and sufficient condition for (1) function to be odd function is that its image is symmetric about the origin; The necessary and sufficient condition for a function to be an even function is that its image is symmetric about y 。
(2) If the domain of a function is symmetric about the origin and the value of the function is always zero, then it is both a odd function and an even function.
(3) If odd function f(x) is meaningful when x=0, then f(0)=0 holds.
(4) If f(x) is an interval monotone function with parity, the monotonicity of odd (even) functions in positive and negative symmetric intervals is the same (inverse).
(5) If the domain of f(x) is symmetric about the origin, then F(x)=f(x)+f(-x) is an even function, and G(x)=f(x)-f(-x) is odd function.
(6) Universal parity
If the function y=f(x) has f(a+x)=f(a-x) for any x in the definition domain, then the image of y=f(x) is symmetric about the straight line x=a, that is, y=f(a+x) is an even function. The function y=f(x) applies to any x in the domain.
Summary of five knowledge points of compulsory mathematics in senior three
The concept of 1. function: Let a and b be non-empty number sets. If any number X in set A has a certain number f(x) corresponding to it according to a certain correspondence F, then F: A → B is called a function from set A to set B. Note: y=f(x), the value of Y corresponding to the value of X. X is called a function value, and the function value set {f(x)|x∈A} is called the range of functions.
Note: If only the analytical formula y=f(x) is given without specifying its domain, the domain of the function refers to the set of real numbers that can make this formula meaningful; The definition and range of functions should be written in the form of sets or intervals.
Domain supplement
The set of real numbers x that can make a function meaningful is called the domain of the function, and the main basis for finding the domain of the function is:
The denominator of (1) score is not equal to zero;
(2) The number of even roots is not less than zero;
(3) The truth value of the logarithmic formula must be greater than zero;
(4) The bases of exponential and logarithmic expressions must be greater than zero and not equal to 1.
(5) If a function is a combination of some basic functions through four operations, then its domain is a set of values of x that make all parts meaningful.
(6) The zero base of the index cannot be equal to zero.
2. The three elements of a function: definition domain, correspondence relationship and value domain.
Note again:
(1) The three elements that make up a function are definition domain, correspondence relationship and value domain. Because the range is determined by the domain and the corresponding relationship, if the domain and the corresponding relationship are completely consistent, the two functions are said to be equal (or the same function).
(2) Two functions are equal if and only if their domains and corresponding relations are completely consistent, regardless of letters representing independent variables and function values. The judgment method of the same function: ① the expressions are the same; (2) Domain consistency (two points must be met at the same time)
Value range supplement
(1), the range of a function depends on the domain and the corresponding laws. No matter what method is adopted to find the range of a function, the domain must be considered first. (2) You should be familiar with the range of linear function, quadratic function, exponential function, logarithmic function and trigonometric function, which is the basis for solving the numerical range of reply. (3) The common methods to find the function range are: direct method and inverse function method.
3. Function image knowledge induction
(1) Definition: In the plane rectangular coordinate system, the set c of points P(x, y) with functions y = f (x) and (x ∈ a) as abscissa and function value y as ordinate is called the image of functions y = f (x) and (x ∈ a).
The coordinates (x, y) of each point on c satisfy the functional relationship y=f(x). On the other hand, the points (x, y) whose coordinates are x and y for each group of ordered real numbers satisfying y=f(x) are all on c, that is, c = {p (x, y) | y = f (x).
Image c is generally a smooth and continuous curve (or straight line), or it may be composed of several curves or discrete points, and it has at most one intersection with any straight line parallel to the Y axis.
(2) Painting
A. Point tracing method: according to the resolution function and the definition domain, find some corresponding values of x and y and list them, trace the corresponding points p (x, y) in the coordinate system with (x, y) as coordinates, and finally connect these points with smooth curves.
B, image transformation method (please refer to the compulsory 4 trigonometric function)
There are three commonly used transformation methods, namely translation transformation, expansion transformation and symmetry transformation.
(3) Function:
1, intuitively see the nature of the function; 2. Analyze the thinking of solving problems by combining numbers and shapes. Improve the speed of solving problems.
Find mistakes in solving problems.
4. Understand the concept of interval.
Classification of (1) interval: open interval, closed interval and semi-open and semi-closed interval; (2) Infinite interval; (3) The number axis representation of the interval.
5. What is mapping?
Generally speaking, let A and B be two nonempty sets. If any element X in set A has an element Y corresponding to it according to a corresponding rule F, the corresponding relation F: AB is called the mapping from set A to set B ... Write it down as "f: ab".
Given a mapping from set a to set b, if A ∈ A, B ∈ B and element a correspond to element b, then we call element b the image of element a and element a the original image of element B.
Note: Function is a special mapping, and mapping is a special correspondence. ① Set A, B and corresponding rule F are definite; (2) The correspondence rule is directional, that is, it emphasizes the correspondence from set A to set B, which is generally different from the correspondence from b to a; ③ For mapping F: A → B, it should be satisfied that: (i) every element in set A has an image in set B, and the image is yes(ii) different elements in set A, and the corresponding images in set B can be the same; (iii) Each element in set B does not need to have an original image in set A. ..
Common function representations and their respective advantages;
Function images can be continuous curves, straight lines, broken lines, discrete points, etc. Pay attention to the basis of judging whether a graph is a function image; Analysis method: the definition domain of the function must be pointed out; Mirror image method: attention should be paid to drawing by tracing point method: determine the definition domain of function; Simplify the analytical formula of the function; Observe the characteristics of the function; List method: the selected independent variables should be representative and reflect the characteristics of the field.
Note: Analytical method: it is convenient to calculate the function value. List method: it is easy to find the function value. Mirror image method: convenient to measure function value
Supplement 1: piecewise function (see textbook P24-25)
There are different functions in different parts of the domain to parse the expression. When finding the function value in different ranges, the independent variable must be substituted into the corresponding expression. The analytic expression of piecewise function cannot be written as several different equations. Instead, write several different expressions of function values and enclose them in left brackets, indicating the values of independent variables of each part respectively. (1) piecewise function is one function, so don't mistake it for several functions. (2) The definition domain of piecewise function is the union of the definition domain of each segment, and the value domain is the union of the value domain of each segment.
Supplement 2: Composite Function
If y=f(u), (u∈M), u=g(x), (x∈A), then y=f[g(x)]=F(x), (x∈A) is called the composite function of f and g.
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